Exercises Chapter 3

8/2/2019 Exercises Chapter 3

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PART IIPRODUCERS, CONSUMERS, AND COMPETITIVE MARKETS

CHAPTER 3CONSUMER BEHAVIOR

REVIEW QUESTIONS1. What does transitivity of preferences mean?

Transitivity of preferences implies that if someone prefers Ato Band prefers Bto C,then he or she prefersAto C.

2. Suppose that a set of indifference curves was not negatively sloped. What could you say about thedesirability of the two goods?

One major assumption of preference theory is that more is preferred to less. Thus, wecan expect that consumers will experience a lower level of satisfaction if we takesome of a good away from them. From this, we necessarily derive negatively slopedindifference curves. However, if one good is undesirable, then less of the undesirablegood leaves the consumer better off, e.g., less toxic waste is preferred to more toxicwaste. When one good is undesirable, the indifference curves showing the trade-offbetween that good and a desired good have positive slopes. In Figure 3.2 below, theindifference curve U2 is preferred to the indifference curve U1.

Figure 3.2

3. Explain why two indifference curves cannot intersect.


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The explanation is most easily achieved with the aid of a graph such as Figure 3.3,which shows two indifference curves intersecting at pointA. We know from thedefinition of an indifference curve that a consumer has the same level of utility alongany given curve. In this case, the consumer is indifferent between bundles Aand Bbecause they both lie on indifference curve U1. Similarly, the consumer is indifferentbetween bundlesAand Cbecause they both lie on indifference curve U2. By the

transitivity of preferences this consumer should also be indifferent between Cand B.However, we see from the graph that Clies above B, so Cmust be preferred to B.Thus, the fact that indifference curves cannot intersect is proven.

Figure 3.3

4. Draw a set of indifference curves for which the marginal rate of substitution is constant. Draw twobudget lines with different slopes; show what the satisfaction-maximizing choice will be in each case.What conclusions can you draw?In Figure 3.4, Good Xand Good Yare perfect substitutes and, thus, the indifferencecurves are straight lines, U1 and U2, each with a slope of -1. For goods that areperfect substitutes, the consumer will always prefer to purchase the cheaper of thetwo goods to obtain maximum utility. For example, if Good Yis cheaper than Good X,the consumer would face the budget constraint L2 and would maximize utility atpointA. On the other hand, if Good Xwere cheaper than Good Y, the consumer wouldface the budget constraint L1 and would maximize utility at point B. If Good XandGood Yhave the same price, the budget constraint would coincide with theindifference curve, and the consumer would be indifferent between any point on the

curve. To see this, recall that the slope of the budget line is More generallly,the slope of a linear indifference curve is the constant rate at which the consumer iswilling to trade the two goods. If the slope of the budget line and the indifferencecurve are the same, then the consumer is indifferent between any point on the budgetline. When the slopes are different, the consumer will choose one of the corners,depending on the respective slopes.


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Figure 3.4

5. Explain why a persons marginal rate of substitution between two goods must equal the ratio ofthe price of the goods for the consumer to achieve maximum satisfaction.The MRSdescribes the rate at which the consumeris willing to trade one good foranother to maintain the same level of satisfaction. The ratio of prices describes thetrade-off that the marketis willing to make between the same two goods. Thetangency of the indifference curve with the budget line represents the point at whichthe trade-offs are equal and consumer satisfaction is maximized. If the MRSbetweentwo goods is not equal to the ratio of prices, then the consumer could trade one goodfor another at market prices to obtain higher levels of satisfaction. This trading

continues until the highest level of satisfaction is achieved.

6. Explain why consumers are likely to be worse off when a product that they consume is rationed.If the maximum quantity of a good is fixed by decree and desired quantities are notavailable for purchase, then there is no guarantee that the highest level ofsatisfaction can be achieved. The consumer will not be able to give up theconsumption of other goods in order to obtain more of the rationed good. Only if theamount rationed is greater than the desired level of consumption can the consumerstill maximize satisfaction without constraint. (Note: rationing may imply a higher

level ofsocialwelfare because of equity or fairness considerations across consumers.)

7. Upon merging with West Germanys economy, East German consumers indicated a preference forMercedes-Benz automobiles over Volkswagen automobiles. However, when they converted theirsavings into deutsche marks, they flocked to Volkswagen dealerships. How can you explain thisapparent paradox?Three assumptions are required to address this question: 1) that a Mercedes costsmore than a Volkswagen; 2) that the East German consumers utility function


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comprises two goods, automobiles and all other goods evaluated in deutsche marks;and 3) that East Germans have incomes. Based on these assumptions, we cansurmise that while once-East German consumers may prefer a Mercedes to aVolkswagen, they either cannot afford a Mercedes or they prefer a bundle of othergoods plus a Volkswagen to a Mercedes alone. While the marginal utility ofconsuming a Mercedes exceeds the marginal utility of consuming a Volkswagon, the

consumer will consider marginal utility per dollar for each good. This means themarginal utility per dollar must have been higher for the Volkswagon sinceconsumers flocked to the Volkswagon dealerships and not the Mercedes dealerships.

8. Describe the equal marginal principle. Explain why this principle may not hold if increasingmarginal utility is associated with the consumption of one or both goods.The equal marginal principle states that the ratio of the marginal utility to pricemust be equal across all goods to obtain maximum satisfaction. This explanationfollows from the same logic examined in Review Question 5. Utility maximization isachieved when the budget is allocated so that the marginal utility per dollar ofexpenditure is the same for each good.

If marginal utility is increasing, the consumer maximizes satisfaction by consumingever larger amounts of the good. Thus, the consumer would spend all income on onegood, assuming a constant price, resulting in a corner solution. With a cornersolution, the equal marginal principle cannothold.

9. What is the difference between ordinal utility and cardinal utility? Explain why the assumption ofcardinal utility is not needed in order to rank consumer choices.Ordinal utility implies an ordering among alternatives without regard for intensity ofpreference. For example, the consumers first choice is preferred to their secondchoice. Cardinal utility implies that the intensity of preferences may be quantified.An ordinal ranking is all that is needed to rank consumer choices. It is not necessary

to know how intensely a consumer prefers basketAover basket B; it is enough toknow thatAis preferred to B.

10. The price of computers has fallen substantially over the past two decades. Use this drop in priceto explain why the Consumer Price Index is likely to substantially understate the cost-of-living indexfor individuals who use computers intensively.The consumer price index measures the changes in the weighted average of theprices of the bundle of goods purchased by consumers. The weights equal the share ofconsumer's expenditures on all of the goods in the bundle. A base year is chosen, andthe weights for that year are used to compute the CPI in that and subsequent years.When the price of a good falls substantially then a consumer will substitute towards

that good, altering the share of that consumer's income spent on each good. By usingthe base year's weights the CPI does not take into account that large price changesalter these expenditure shares, and so gives an inaccurate measure of changes in thecost of living.

For example, assume Fred spends 10% of his income on computers in 1970, and thatFred's expenditure shares in 1970 were used as the weights to calculate Fred's CPI insubsequent years. If Fred's demand for computers was inelastic, then reductions inthe price of computers (relative to other goods) would reduce the share of his income


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spent on computers. After 1970 a CPI that used Fred's 1970 expenditure shares asweights would give a 10% weight to the falling price of computers, even though Fredspent less that 10% of his income on computers. As long as the prices of other goodsrose, or fell less than 10%, then the CPI gives too little weight to the changes in theprices of other goods, and understates the changes in Fred's cost of living.

EXERCISES1. In this chapter, consumer preferences for various commodities did not change during the analysis.Yet in some situations, preferences do change as consumption occurs. Discuss why and howpreferences might change over time with consumption of these two commodities:

a. cigarettesThe assumption that preferences do not change is a reasonable one if choices areindependent across time. It does not hold, however, when "habit-forming" oraddictive behavior is involved, as in the case of cigarettes: the consumption ofcigarettes in one period influences their consumption in the next period.

b. dinner for the first time at a restaurant with a special cuisineWhile there may not be anything physically addictive in dining at new and differentrestaurants, one can become better informed about a particular restaurant. One mayenjoy choosing more new and different restaurants, or one may be tired of choosinganother new and different place to eat. In either of these cases, choices change asconsumption occurs.

2. Draw the indifference curves for the following individuals preferences for two goods: hamburgersand beer.a. Al likes beer but hates hamburgers. He always prefers more beer no matter howmany hamburgers he has.For Al, hamburgers are a "bad." His indifference curves slope upward and to theright rather than downward and to the left. For Al, U1 is preferred to U2 and U2 ispreferred to U3. See figure 3.2a. If you instead assumed that hamburgers were aneutral good, then the indifference curves would be vertical and utility is increasingto the right as more beer is consumed.

b. Betty is indifferent between bundles of either three beers or two hamburgers. Herpreferences do not change as she consumes any more of either food.

Since Betty is indifferent between three beers and two burgers, an indifference curveconnects these two points. Bettys indifference curves are a series of parallel lines

with slope of . See figure 3.2b.

c. Chris eats one hamburger and washes it down with one beer. He will not consumean additional unit of one item without an additional unit of the other.


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For Chris, hamburgers and beer are perfect complements, i.e., he always wants toconsume the goods in fixed proportions to each other. The indifference curves are L-shaped, with corners on a 45-degree line out of the origin. See figure 3.2c.

Figure 3.2.a

Figure 3.2.b


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d. Doreen loves beer but is allergic to beef. Every time she eats a hamburger shebreaks out in hives.For Doreen, hamburgers are not considered a "good" but rather a "bad," and thus herpreferred position is not upwards and to the right, but rather downward and to theright. For Doreen, U1 is preferred to U2 and U2 is preferred to U3. See figure 3.2d.

Figure 3.2.c

Figure 3.2.d

3. The price of tapes is $10 and the price of CDs is $15. Philip has a budget of $100 and has alreadypurchased 3 tapes. He thus has $70 more to spend on additional tapes and CDs. Draw his budget


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line. If his remaining expenditure is made on 1 tape and 4 CDs, show Philips consumption choice onthe budget line.Given Philips remaining income of $70, he can afford 7 tapes if he spends the entireamount on tapes, and he can afford 4.7 CDs if he spends the entire amount on CDs.According to figure 3.3, his budget line therefore intersects the vertical axis at a

quantity of 7 tapes and the horizontal axis at a quantity of 4.7 CDs. Since he facesconstant prices, the budget line has a constant slope and is a straight line.

Figure 3.3

4. Debra usually buys a soft drink when she goes to a movie theater, where she has a choice of threesizes: the 8 ounce drink costs $1.50, the 12 ounce drink, $2.00, and the 16 ounce drink, $2.25.Describe the budget constraint that Debra faces when deciding how many ounces of the drink topurchase. (Assume that Debra can costlessly dispose of any of the soft drink that she does not want.

First notice that as the size of the drink increases, the price eper ounce decreases.

When she buys the 8 ounce soft drink she pays When shebuys the 12 ounce size she pays $0.17 per ounce, and when she buys the 16 ouncesize, she pays $0.14 per ounce. Given that there are three different prices per ounceof soft drink, the budget line will have two kinks in it, as illustrated in figure 3.4.


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Figure 3.4

5. Suppose Bill views butter and margarine as perfectly substitutable for each other.a. Draw a set of indifference curves that describes Bills preferences for butter andmargarine.

Figure 3.5.a


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b. Are these indifference curves convex? Why?Convexity implies that a line segment connecting any two points on a curve liesabove the curve, i.e., the curve is "bowed inward." Because the consumer considersbutter and margarine to be perfect substitutes, there is no diminishing marginalutility, and the resultant indifference curves are straight lines. Straight-line

indifference curves are not strictly convex.

c. If butter costs $2 per package, while margarine costs only $1, and Bill has a $20budget to spend for the month, which butter-margarine market basket will hechoose? Can you show this graphically?Let Bills income be represented by Y, the price of butter by PB, the quantity of butterby B, the price of margarine by PM, and the quantity of margarine by M. Then thegeneral form of the budget constraint is:

Y = PBB + PMM.

Substituting for the given values ofY, PB, and PM, we obtain the specificrepresentation of Bills budget constraint:

20 = 2B+ 1M, or B= 10 - 0.5M.

Because Bill is indifferent between butter and margarine, and the price of butter isgreater than the price of margarine, Bill will only buy margarine. This is a cornersolution, because the optimal choice occurs on an axis. In Figure 3.5.c Bills utilitymaximizing bundle is pointA.

Figure 3.5.c

6. Suppose that Jones and Smith have decided to allocate $1,000 per year on liquid refreshments inthe form of alcoholic or nonalcoholic drinks. Jones and Smith differ substantially in their preferences


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for these two forms of refreshment. Jones prefers alcoholic to nonalcoholic drinks, while Smithprefers the nonalcoholic option.a. Draw a set of indifference curves for Jones and a second set for Smith.

Figure 3.6.a

b. Discuss why the two sets of curves are different from each other using the conceptof marginal rate of substitution.At any combination of alcoholic,A, and nonalcoholic, N, drinks, Jones is willing togive up a lot ofNfor an additional unit ofA; whereas, Smith is willing to give up a lot

ofAfor an additional unit ofN. Because Jones needs more Nto compensate him forgiving up someA, he has a lower marginal rate of substitution of alcoholic fornonalcoholic beverages than Smith. With alcoholic beverages on the vertical axis,Jones indifference curves are less steep than Smiths at any point on the graph.

c. If both Smith and Jones pay the same prices for their refreshments, will theirmarginal rates of substitution of alcoholic for nonalcoholic drinks be the same ordifferent? Explain.In order to maximize utility, the consumer must consume quantities such that theMRSbetween any two commodities is equal to the ratio of prices. If Smith and Jonesare rational consumers, their MRSmust be equal because they face the same market

prices. But because they have different preferences, they will consume differentamounts of the two goods, alcoholic and nonalcoholic. At those different levels,however, their MRSare equal.

7. Consumers in Georgia pay twice as much for avocados as they do for peaches. However, avocadosand peaches are equally priced in California. If consumers in both states maximize utility, will themarginal rage of substitution of peaches for avocados be the same for consumers in both states? Ifnot, which will be higher?


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The marginal rate of substitution of peaches for avocados is the amount of avocadosthat a person is willing to give up to obtain one additional peach. When consumersmaximize utility, they set their marginal rate of substitution equal to the price ratio,

which in this case is In Georgia, , which means that when

consumers are maximizing utility, In California,

, which means that when consumers are maximizing utility, Themarginal rate of substitution is therefore not the same in both states, and will behigher in California.

8. Anne is a frequent flyer whose fares are reduced (through coupon giveaways) by 25 percent aftershe flies 25,000 miles a year, and then by 50 percent after she flies 50,000 miles. Can you graph thebudget line that Anne faces in making her flight plans for the year?In Figure 3.8, we plot miles flown, M, against all other goods, G, in dollars. Thebudget constraint is:

Y = PMM + PGG, or

The slope of the budget line is . In this case, the price of miles flown changes asthe number of miles flown changes, so the budget curve is kinked at 25,000 and at50,000 miles. If we assume PM is $1 per mile for less than or equal to 25,000 miles,then PM= $0.75 for 25,000 < M 50,000, and PM= $0.50 for M> 50,000. Also,assume that PG= $1.00. The slope of the budget line from Ato Bis -1, the slope of thebudget line from Bto Cis -0.75, and the slope of the budget line from Cto Dis -0.5.


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Figure 3.8

9. Antonio buys 8 new college textbooks during his first year at school at a cost of $50 each. Usedbooks cost only $30 each. When the bookstore announces that there will be a 20 percent priceincrease in new texts and a 10 percent increase in used texts for the next year, Antonios fatheroffers him $80 extra. Is Antonio better off or worse off after the price change?It follows from the axiom of revealed preferences that, since Antonio chose to

purchase only new textbooks when both new and used textbooks were available, itmust be the case that Antonio does not consider used textbooks substitutes for newtextbooks at the old prices.

With the increase in price, however, to $60 for new textbooks and $33 for used texts,

the relative price of new texts to used texts increases from to .Antonio may react to the relative price increase in one of two ways:

(1) If new and used texts are notsubstitutes for Antonio (L-shaped indifferencecurves), then Antonio will be just as well off when the price of new texts rises and hisfather gives him $80 (= (60 - 50)8).

(2) If he chooses to buy a few used texts in response to the relative price increase(given the extra $80), he will move to a higher indifference curve and will thereforebe better off. See Figures 3.9.a and 3.9.b.


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Figure 3.9.a

Figure 3.9.b

10. Suppose that Samantha and Jason both spend $24 per week on video and movie entertainment.When the prices of videos and movies are both $4, they both rent 3 videos and buy 3 movie tickets.Following a video price war and an increased cost of movie tickets, the video price falls to $2 and themovie ticket increases to $6. Samantha now rents 6 videos and buys 2 movie tickets; Jason, however,buys 1 movie ticket and rents 9 videos.a. Is Samantha better off or worse off after the price change?Samanthas original point of utility maximization may be represented by point AonU1 in Figure 3.10.a. With the new prices, Samantha could still afford to choose


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bundleA: $24 = $2(3 videos) + $6 (3 movies). The fact that she chose bundle Brevealsshe has obtained a higher level of utility, U2. See Figure 3.10.a.

Figure 3.10.a

b. Is Jason better off or worse off?Similarly, Jason must also be better off.

11. Connie Consumer has a monthly income of $200 which she allocates among two goods: meat andpotatoes.a. Suppose meat costs $4 per pound and potatoes cost $2 per pound. Draw her budgetconstraint.Let M= meat and P= potatoes. Connies budget constraint is

$200 = 4M+ 2P, or

M= 50 - 0.5P.

As shown in Figure 3.11.a, with Mon the vertical axis, the vertical intercept is 50.The horizontal intercept may be found by setting M= 0 and solving for P.


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Figure 3.11.a

b. Suppose also that her utility function is given by the equation u(M, P) = 2M + P.What combination of meat and potatoes should she buy to maximize her utility?(Hint: Meat and potatoes are perfect substitutes.)Connies utility is equal to 100 when she buys 50 pounds of meat and no potatoes orno meat and 100 pounds of potatoes. The indifference curve for U= 100 coincideswith her budget constraint. Any combination of meat and potatoes along this line willprovide her with maximum utility.

c. Connies supermarket has a special promotion. If she buys 20 pounds of potatoes(at $2 per pound), she gets the next 10 pounds for free. This offer applies only to thefirst 20 pounds she buys. All potatoes in excess of the first 20 pounds (excludingbonus potatoes) are still $2 per pound. Draw her budget constraint.Figure 3.11.c represents Connies budget constraint when the supermarket runs itsspecial promotion on potatoes. Notice that her budget constraint has a slope of -2until Connie has purchased twenty pounds of potatoes, is then flat, since the ten nextpounds of potatoes are free, and then has a slope of -2 until it intercepts the potatoaxis at 110.


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Figure 3.11.c

d. An outbreak of potato rot raises the price of potatoes to $4 per pound. Thesupermarket ends its promotion. What does her budget constraint look like now?What combination of meat and potatoes maximizes her utility?With the price of potatoes at $4, Connie may buy either 50 pounds of meat or 50pounds of potatoes, or some combination in between. See Figure 3.11.d. Shemaximizes utility at U= 100 at pointAwhen she consumes 50 pounds of meat andno potatoes. This is a corner solution.

Figure 3.11.d


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12. The utility that Jane receives by consuming food F and clothing C is given by:u(F,C) = FC.a. Draw the indifference curve associated with a utility level of 12 and theindifference curve associated with a utility level of 24. Are the indifference curvesconvex?To find the baskets of food, F, and clothing, C, which yield satisfactions of 12 and 24

solve the equations and .

U= 12 U= 24

Food Clothing Food Clothing

1.0 12.0 1.0 24.0

1.5 8.0 2.0 12.0

2.0 6.0 3.0 8.0

3.0 4.0 4.0 6.0

4.0 3.0 6.0 4.0

6.0 2.0 8.0 3.0

8.0 1.5 12.0 2.0

12.0 1.0 24.0 1.0

The indifference curves are convex.

b. Suppose that food costs $1 a unit, clothing costs $3 a unit, and Jane has $12 tospend on food and clothing. Graph the budget line that she faces.The budget constraint is:

Y = PFF + PCC, or

12 = 1F+ 3C, or .

See Figure 3.12.a.


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Figure 3.12.a

c. What is the utility-maximizing choice of food and clothing? (Hint: Solve theproblem graphically.)The highest level of satisfaction occurs where the budget line is tangent to thehighest indifference curve. In Figure 3.12.a this is at the point F= 6 and C= 2. Tocheck this answer, note that it exhausts Janes income, 12 = 6PF+ 2PC. Also, thisbundle yields a satisfaction of 12, as (6)(2) = 12. See Figure 3.12.a.

d. What is the marginal rate of substitution of food for clothing when utility ismaximized?At the utility-maximizing level of consumption, the slope of the indifference curve isequal to the slope of the budget constraint. Since the MRSis equal to the negativeslope of the indifference curve, the MRSin this problem is equal to one-third. Thus,Jane would be willing to give up one-third of a unit of clothing for one unit of food.

e. Suppose that Jane buys 3 units of food and 3 units of clothing with her $12 budget.Would her marginal rate of substitution of food for clothing be greater or less than1/3? Explain.If Jane buys 3 units of food for $1.00 per unit and 3 units of clothing for $3.00 perunit, she would spend all her income. However, she would obtain a level of

satisfaction of only 9, which represents a sub-optimal choice. At this point, the MRSis greater than one-third, and thus, at the prices she faces, she would welcome theopportunity to give up clothing to get more food. She is willing to trade clothing forfood until her MRSis equal to the ratio of prices. See Figure 3.12.e.


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Figure 3.12.e

13. The utility that Meredith receives by consuming food F and clothing C is given by u(F,C) = FC.Suppose that Merediths income in 1990 is $1,200 and the prices of food and clothing are $1 per unitfor each. However, by 1995 the price of food has increased to $2 and the price of clothing to $3. Let100 represent the cost of living index for 1990. Calculate the ideal and the Laspeyres cost-of-livingindex for Meredith for 1995. (Hint: Meredith will spend equal amounts on food and clothing withthese preferences.)Laspeyres Index

The Laspeyres index represents how much more Meredith would have to spend in

1995 versus 1990 if she consumed the same amounts of food and clothing in 1995 asshe did in 1990. That is, the Laspeyres index for 1995 (L) is given by:

L = 100 (Y )/Y

where Y represents the amount Meredith would spend at 1995 prices consuming thesame amount of food and clothing as in 1990: Y = PFF + PCC = 2F + 3C, where Fand C represent the amounts of food and clothing consumed in 1990.

We thus need to calculate F and C, which make up the bundle of food and clothingwhich maximizes Merediths utility given 1990 prices and her income in 1990. Usethe hint to simplify the problem: Since she spends equal amounts on both goods, P FF

= PCC. Or, you can derive this same equation mathematically: With this utilityfunction, MUC =U/ C = F, and MUF = U/ F = C. To maximize utility, Meredithchooses a consumption bundle such that MUF/MUC = PF/PC, which again yields PFF =PCC.

From the budget constraint, we also know that:

PF F +PC C = Y.


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Combining these two equations and substituting the values for the 1990 prices andincome yields the system of equations:

C = F and C + F = 1,200.

Solving these two equations, we find that:

C = 600 and F = 600.

Therefore, the Laspeyres cost-of-living index is:

L = 100(2F + 3C)/Y = 100[(2)(600) + (3)(600)]/1200 = 250.

Ideal IndexThe ideal index represents how much more Meredith would have to spend in 1995versus 1990 if she consumed amounts of food and clothing in 1995 which would giveher the same amount of utility as she had in 1990. That is, the ideal index for 1995

(I) is given by:

I = 100(Y'')/Y, where Y'' = P'FF + P'CC' = 2F' + 3C'

where F' and C' are the amount of food and clothing which give Meredith the sameutility as she had in 1990. F' and C' must also be such that Meredith spends the leastamount of money at 1995 prices to attain the 1990 utility level.

The bundle (F',C') will be on the same indifference curve as (F,C) and the indifferencecurve at this point will be tangent to a budget line with slope -(P'F/P'C), where P'F andP'C are the prices of food and clothing in 1995. Since Meredith spends equal amountson the two goods, we know that 2F' = 3C'. Since this bundle lies on the same

indifference curve as the bundle F = 600, C = 600, we also know that F'C' =(600)(600).

Solving for F' yields:

F'[(2/3)F'] = 360,000 or F' = = 734.8.

From this, we obtain C':

C' = (2/3)F' = (2/3)734.8 = 489.9.

We can now calculate the ideal index:

I = 100(2F' + 3C')/Y = 100[2(734.8) + (3)(489.9)]/1200 = 244.9.

Exercises Chapter 3

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